Stage-wise adaptive designs

Stage-Wise Adaptive Designs presents the theory and methodology of stage-wise adaptive design across various areas of study within the field of statistics, from sampling surveys and time series analysis to generalized linear models and decision theory. Providing the necessary background material along with illustrative S-PLUS functions, this book serves as a valuable introduction to the problems of adaptive designs.

The author begins with a cohesive introduction to the subject and goes on to concentrate on generalized linear models, followed by stage-wise sampling procedures in sampling surveys. Adaptive forecasting in the area of time series analysis is presented in detail, and two chapters are devoted to applications in clinical trials. Bandits problems are also given a thorough treatment along with sequential detection of change-points, sequential applications in industrial statistics, and software reliability.

S-Plus functions are available to accompany particular computations, and all examples can be worked out using R, which is available on the book's related FTP site. In addition, a detailed appendix outlines the use of these software functions, while an extensive bibliography directs readers to further research on the subject matter.

Assuming only a basic background in statistical topics, Stage-Wise Adaptive Designs is an excellent supplement to statistics courses at the upper-undergraduate and graduate levels. It also serves as a valuable reference for researchers and practitioners in the fields of statistics and biostatistics.

Author Notes

Shelemyahu Zacks, PhD, is Professor of Statistics in the Department of Mathematical Sciences at Binghamton University. He has published several books and over 170 journal articles in the areas of design and analysis of experiments, statistical control of stochastic processes, statistical decision theory, statistical methods in logistics, and sampling from finite populations. Dr. Zacks is a Fellow of the American Statistical Association, Institute of Mathematical Sciences, and American Association for the Advancement of Sciences.

Preface	p. xiii
1 Synopsis	p. 1
1.1 Multistage and Sequential Estimation	p. 1
1.2 Adaptive Designs for Generalized Linear Models	p. 3
1.3 Adaptive Methods for Sampling from Finite Populations	p. 5
1.4 Adaptive Prediction and Forecasting in Time Series Analysis	p. 6
1.5 Adaptive Search of an MTD in Cancer Phase I Clinical Trials	p. 7
1.6 Adaptive and Sequential Procedures in Phase III Clinical Trials	p. 9
1.7 Sequential Allocation of Resources	p. 10
1.8 Sequential Detection of Change Points	p. 12
1.9 Sequential Methods in Industrial Testing, Reliability, and Design of Experiments	p. 13
2 Multistage and Sequential Estimation	p. 15
2.1 Stein's Two-Stage Procedure	p. 15
2.2 Modifications to Attain Asymptotic Efficiency	p. 18
2.3 Two-Stage Sampling from Exponential Distributions	p. 20
2.3.1 Fixed-Width Confidence Interval for the Location Parameter of an Exponential Distribution	p. 20
2.3.2 Two-Stage Sampling for a Bounded Risk Point Estimation of the Exponential Parameter	p. 24
2.4 Sequential Fixed-Width Interval Estimation	p. 34
2.5 Distributions of Stopping Variables of Sequential Sampling	p. 37
2.5.1 General Theory	p. 38
2.5.2 Characteristics of Ray's Procedure	p. 40
2.5.3 Risk of Some Sequential Point Estimators	p. 41
2.6 Sequential Fixed-Width Intervals for the Log-Odds in Bernoulli Trials	p. 42
2.6.1 Problem	p. 42
2.6.2 Distribution of N (¿)	p. 43
2.6.3 Functionals of ¿&hat;N(¿)	p. 47
2.7 Bayesian Sequential Estimation	p. 49
2.7.1 General Theory	p. 49
2.7.2 Estimating the Scale Parameter of the Exponential Distribution	p. 51
3 Adaptive Designs for Generalized Linear Models	p. 55
3.1 Exponential Example	p. 55
3.2 Adaptive Designs for the Fisher Information	p. 57
3.3 Adaptive Bayesian Designs	p. 63
3.4 Adaptive Designs for Inverse Regression	p. 66
3.4.1 Non-Bayesian Adaptive Designs	p. 66
3.4.2 Bayesian Adaptive Designs, ¿ Known	p. 71
3.5 Stochastic Approximation	p. 73
4 Adaptive Methods for Sampling from Finite Populations	p. 75
4.1 Basic Theory	p. 75
4.1.1 Design Approach	p. 76
4.1.2 Modeling Approach	p. 79
4.2 Two-Stage and Sequential Estimation of the Population Mean	p. 81
4.2.1 Design Approach: SRSWR	p. 81
4.2.2 Design Approach: SRSWOR	p. 84
4.2.3 Modeling Approach	p. 86
4.3 Adaptive Allocation of Stratified SRS	p. 86
4.3.1 Basic Theory	p. 87
4.3.2 Two-Stage Procedure for a Fixed-Width Interval Estimation of &Ybar;n Under Stratified Sampling	p. 88
4.4 Adaptive Search for Special Units	p. 91
4.5 Adaptive Estimation of the Size of a Finite Population	p. 92
4.6 Applications in Software Reliability	p. 96
4.6.1 Sequential Stopping for Time Domain Models	p. 96
4.6.2 Sequential Stopping for Data Domain Models	p. 98
4.7 Sampling Inspection Schemes	p. 100
4.7.1 Two-Stage Sampling for Attributes	p. 100
4.7.2 Sequential Sampling for Attributes	p. 102
4.8 Dynamic Bayesian Prediction	p. 103
5 Adaptive Prediction and Forecasting in Time Series Analysis	p. 107
5.1 Basic Tools of Time Series Analysis	p. 107
5.2 Linear Predictors for Covariance Stationary T.S.	p. 113
5.2.1 Optimal Linear Predictors	p. 113
5.2.2 Minimal PMSE Predictors for AR(p) T.S.	p. 117
5.2.3 Prediction with Unknown Covariance Structure	p. 119
5.2.4 ARIMA Forecasting	p. 121
5.3 Quadratic LSE Predictors for Nonstationary T.S.	p. 124
5.4 Moving Average Predictors for Nonstationary T.S.	p. 128
5.4.1 Linear MAS Predictors	p. 130
5.5 Predictors for General Trends with Exponential Discounting	p. 132
5.5.1 Recursive Computations with Shifted Origin	p. 133
5.5.2 Linear Trend	p. 136
5.5.3 Linear Trend with Cyclical Components	p. 137
5.6 Dynamic Linear Models	p. 141
5.6.1 Recursive Computations for the Normal Random Walk DLM, Constant Variances	p. 143
5.6.2 Incorporating External Information	p. 146
5.6.3 General DLM with Applications	p. 147
5.6.4 DLM for ARMA(p,q) T.S.	p. 153
5.7 Asymptotic Behavior of DLM	p. 157
5.8 Linear Control of DLM	p. 163
5.8.1 Deterministic Linear Control	p. 163
5.8.2 Stochastic Linear Control	p. 166
6 Adaptive Search of an MTD in Cancer Phase I Clinical Trials	p. 171
6.1 Up-and-Down Adaptive Designs	p. 171
6.2 Bayesian Adaptive Search: The Continuous Reassessment Method	p. 177
6.3 Efficient Dose Escalation with Overdose Control	p. 179
6.4 Patient-Specific Dosing	p. 181
6.5 Toxicity versus Efficacy	p. 182
7 Adaptive and Sequential Procedures in Clinical Trials, Phases II and III	p. 185
7.1 Randomization in Clinical Trials	p. 185
7.2 Adaptive Randomization Procedures	p. 187
7.2.1 Random Allocation Rule	p. 187
7.2.2 Truncated Binomial Design	p. 189
7.2.3 Efron's Biased Coin Design	p. 190
7.2.4 Wei's Urn Design	p. 192
7.2.5 Response Adaptive Designs	p. 192
7.3 Fixed-Width Sequential Estimation of the Success Probability in Bernoulli Trials	p. 193
7.4 Sequential Procedure for Estimating the Probability of Success in Bernoulli Trials	p. 197
7.5 Sequential Comparison of Success Probabilities	p. 198
7.6 Group Sequential Methods	p. 200
7.7 Dynamic Determination of Stage-Wise Sample Size	p. 205
7.7.1 Truncated Three-Stage Procedure for Power (TTSP)	p. 205
7.7.2 Bartoff-Lai GLR Procedure	p. 208
8 Sequential Allocation of Resources	p. 211
8.1 Bernoulli Bandits	p. 211
8.2 Gittins Dynamic Allocation Indices	p. 216
8.3 Sequential Allocations in Clinical Trials	p. 218
8.4 Bernoulli Bandits with Change Point	p. 221
8.4.1 Introduction	p. 221
8.4.2 Optimizing the Final Cycle	p. 222
8.4.3 Surveillance Cycle	p. 224
8.4.4 Multiple Surveillance Cycles	p. 230
8.5 Sequential Designs for Estimating the Common Mean of Two Normal Distributions: One Variance Known	p. 233
9 Sequential Detection of Change Points	p. 237
9.1 Bayesian Detection When the Distributions Before and After the Change Are Known	p. 237
9.1.1 Problem	p. 237
9.1.2 Bayesian Framework	p. 238
9.1.3 Application in System Reliability	p. 241
9.1.4 Optimal Stopping for Detecting a Change Point in the intensity of a Poisson Process	p. 243
9.2 Bayesian Detection When the Distributions Before and After the Change Are Unknown	p. 245
9.2.1 Bayesian Framework	p. 246
9.2.2 Optimal Stopping Rules	p. 247
9.2.3 Detecting a Change in the Success Probabilities of Binomial Trials: An Example	p. 248
9.3 CUSUM Procedures for Sequential Detection	p. 250
9.3.1 Structure of CUSUM Procedures	p. 250
9.3.2 Asymptotic Minimaxity of CUSUM Procedures	p. 251
9.3.3 Exact Distribution of Stopping Variables in CUSUM Procedures	p. 253
9.4 Tracking Algorithms for Processes with Change Points	p. 258
9.4.1 General Comments	p. 258
9.4.2 Tracking a Process with Change Points	p. 259
9.4.3 Recursive Nonlinear Estimation with Moving Windows	p. 260
9.4.4 Specific Cases	p. 263
9.4.5 Case Studies	p. 269
9.5 Recursive Estimation with Change Points	p. 272
9.5.1 Reaction of Recursive Estimators to Change Points	p. 272
9.5.2 Recursive Estimation with the Kalman Filter	p. 276
9.5.3 Detecting Change Points in Recursive Estimation	p. 278
9.5.4 Adjustment of the Kalman Filter for Change Points	p. 279
9.5.5 Special Case	p. 282
9.6 Additional Theoretical Contributions	p. 283
10 Sequential Methods In Industrial Testing	p. 285
10.1 Sequential Testing (SPRT)	p. 285
10.1.1 Wald Sequential Probability Ratio Test	p. 286
10.1.2 Exact Distributions of N in the Exponential Case	p. 294
10.2 Characteristics of Sequential Procedures in Reliability Estimation and Testing	p. 298
10.2.1 Reliability Estimation	p. 299
10.2.2 Reliability Testing	p. 302
10.2.3 Total Operating Time of Repairable System	p. 304
10.3 Some Comments on Sequential Design of Experiments	p. 305
10.4 Sequential Testing of Software Reliability	p. 308
10.4.1 Complete Bayesian Model	p. 309
10.4.2 Empirical Bayes: Adaptive Approach When ¿ and ¿ Are Unknown	p. 310
Bibliography	p. 313
Appendix: SPLUS/R Programs	p. 336
Author Index	p. 375
Topic Index	p. 382

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Table of Contents